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Physical Interpretation of Relationship Between Hall Conductivity and Berry Curvature?

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Why is the Hall conductivity in a 2D material $$\tag{1} \sigma_{xy}=\frac{e^2}{2\pi h} \int dk_x dk_y F_{xy}(k)$$ where the integral is taken over the Brillouin Zone and $F_{xy}(k)$ is the Berry curvature of the filled bands? What is the physical interpretation of this equation?

Also, can we re-parametrize all of the filled states by another pair of variables $A$ and $B$ and conclude that $$\tag{2} \sigma_{xy}=\frac{e^2}{2\pi h} \int F(A,B)dAdB$$ where $F(A,B)$ is the Berry curvature with respect to the $A$ and $B$ parameter space?

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user ChickenGod
asked Nov 13, 2013 in Theoretical Physics by ChickenGod (70 points) [ no revision ]

1 Answer

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The formula follows from the Kubo formula of conductivity (based on the linear response theory), which is discussed in this question: Kubo Formula for Quantum Hall Effect and in the references therein. Starting from the Kubo formula (set $e=\hbar=1$) $$\tag{1}\sigma_{xy}=i\sum_{E_m<0<E_n}\frac{\langle m|v_x|n\rangle\langle n|v_y|m\rangle-\langle m|v_y|n\rangle\langle n|v_x|m\rangle}{(E_m-E_n)^2},$$ where $|m\rangle$ is the single particle eigen state of the eigen energy $E_m$, i.e. $$\tag{2} H|m\rangle = E_m|m\rangle.$$ Let us take the momentum derivative $\partial_k$ on both sides of Eq. (2), we have $$\tag{3}(\partial_{k}H)|m\rangle + H\partial_{k}|m\rangle = (\partial_{k}E_m)|m\rangle + E_m \partial_{k}|m\rangle.$$ Then overlap with $\langle n|$ from left, Eq. (3) becomes $$\tag{4}\langle n|(\partial_{k}H)|m\rangle + E_n\langle n|\partial_{k}|m\rangle = (\partial_{k}E_m)\langle n|m\rangle + E_m \langle n|\partial_{k}|m\rangle.$$ Here we have used $\langle n|H = E_n\langle n|$. If $|m\rangle$ and $|n\rangle$ are different eigen states (for $E_m\neq E_n$ in Eq. (1)), their overlap should vanish, i.e. $\langle n|m\rangle=0$. Also note that $\partial_k H$ is nothing but the velocity operator $v=\partial_k H$ by definition. So Eq. (4) can be reduced to $$\tag{5} \langle n|v|m\rangle = (E_m - E_n) \langle n|\partial_{k}|m\rangle.$$ Substitute Eq. (5) to Eq. (1) (restoring the $x$, $y$ subscript), we have $$\tag{6} \sigma_{xy}=-i\sum_{E_m<0<E_n}\big(\langle m|\partial_{k_x}|n\rangle\langle n|\partial_{k_y}|m\rangle - \langle m|\partial_{k_y}|n\rangle\langle n|\partial_{k_x}|m\rangle\big).$$

On the other hand, the Berry connection is defined as $A_\mu=i\langle m|\partial_{k_\mu}|m\rangle$, and the Berry curvature is $F_{xy}=\partial_{k_x}A_{y}-\partial_{k_y}A_{x}$. Given that $(\partial_k\langle m|)|n\rangle = - \langle m|\partial_k|n\rangle$ (integrate by part), we can see $$\tag{7} F_{xy}= -i \sum_n\big( \langle m|\partial_{k_x}|n\rangle\langle n|\partial_{k_y}|m\rangle - \langle m|\partial_{k_y}|n\rangle\langle n|\partial_{k_x}|m\rangle\big) +i \langle m|\partial_{k_x}\partial_{k_y}-\partial_{k_y}\partial_{k_x}|m\rangle.$$ The last term will vanish as the partial derivatives commute with each other. So, by comparing with Eq. (6), we end up with $$\tag{8} \sigma_{xy}=\sum_{E_m<0}F_{xy}\sim\int_{BZ} d^2k F_{xy}.$$ This means the Hall conductance is simply the sum of the Chern numbers (the total Berry flux through the BZ) for all the occupied bands.

So what is the physical meaning of $F_{xy}$? $F_{xy}$ is an effective magnetic field in the momentum space. We know that for the magnetic field $B$ in the real space, a charged particle moving in it will experience the Lorentz force, such that the equation of motion reads $\dot{k}=\dot{r}\times B$. Now to switch to the momentum space, we just need to exchange the momentum $k$ and the coordinate $r$, and replace $B$ by $F$, which leads to $$\tag{9} \dot{r}=\dot{k}\times F$$ So what is $\dot{r}$? It is the velocity of the electron, which is proportional to the current $j$. And what is $\dot{k}$? It is the force acting on the electron, which is proportional to the electric field strength $E$, so Eq. (9) implies $$\tag{10} j \sim E\times F.$$ Therefore the Berry curvature $F_{xy}$ simply gives the Hall response of each single electron state. So the Hall conductivity of the whole electron system should be the sum of the Berry curvature over all occupied states, which is stated in Eq. (8).

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user Everett You
answered Nov 16, 2013 by Everett You (660 points) [ no revision ]
A fantastic answer! Although I was looking for pictorial intuition, your equations (9) and (10) were a beautiful way at looking at the relationship between conductivity and Berry curvature. However, where do the edge states enter in this formalism? And also, I've never seen the Kubo formula expressed as your equation (1). Finally, even though the Kubo formula is an approximation, your equation (6) should be exact by Laughlin's Argument, right?

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user ChickenGod
@ChickenGod The edge state does not enter in this formalism. The Kubo formula describes the bulk response. All the information needed to calculate the conductivity is the single particle wave functions in the bulk. Presumably the wave function is subject to periodic boundary conditions, so there is no edge in this formalism at all. But the amazing thing is that there is this bulk-boundary duality, that the bulk response can be encoded in the boundary. But that is another long story then.

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user Everett You
Kubo formula is indeed an approximation, so as Eq. (6), which is formulated in terms of free fermions. The interaction and disorder effects are not included. So the amazing thing is that even with electron interaction, the Hall conductivity is still quantized exactly to the same integer value, which was shown by Laughlin's argument. The point is that for gapped electron system, the Hall conductance is a topological character, which is invariant under any deformation as long as the gap is not closed. So the quantization is always exact, which is beyond the description of Kubo formula.

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user Everett You
@Everett You : The fractional quantum hall effect is a counter example.

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user jjcale
@jjcale Do you mean FQH is a counter example of the bulk-boundary correspondence or a counter example of the topological robustness?

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user Everett You
@Everett You : It's a counter example to the quantization to the integer value and to Laughlin's Argument.

This post imported from StackExchange Physics at 2014-03-24 04:13 (UCT), posted by SE-user jjcale

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